Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University

Gabriel Kotliar

Physics Department and

Center for Materials Theory

Rutgers University

THE STATE UNIVERSITY OF NEW JERSEY

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The Strong Correlation Problem

Two limiting cases of the electronic structure of solids are understood:the high density limit and the limit of well separated atoms.

High densities, the is electron be a wave, use band theory, k-space

One particle excitations: quasi-particle,quasi-hole bands, collective modes.

Density Functional Theory with approximations suggested by the Kohn Sham formulation, (LDA GGA) is a successful computational tool for the total energy, and a good starting point for perturbative calculation of spectra, GW.……………………


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Mott : Correlations localize the electron

Low densities, electron behaves as a particle,use atomic physics, real space

One particle excitations: Hubbard Atoms: sharp excitation lines corresponding to adding or removing electrons. In solids they broaden by their incoherent motion, Hubbard bands (eg. bandsNiO, CoO MnO….)

Magnetic and Orbital Ordering at low T

Quantitative calculations of Hubbard bands and exchange constants, LDA+ U, Hartree Fock. Atomic Physics.


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Localization vs Delocalization Strong Correlation Problem

•A large number of compounds with electrons which are not close to the well understood limits (localized or itinerant).•These systems display anomalous behavior (departure from the standard model of solids).•Neither LDA or LDA+U or Hartree Fock works well•Dynamical Mean Field Theory: Simplest approach to the electronic structure, which interpolates correctly between atoms and bands


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Outline

Motivation: plutonium puzzles. Review of Dynamical Mean Field

Theory an Extension to realistic systems . DMFT and DFT.

A case study of system specific properties: f .electrons DMFT Results for Pu.

A case study of system specific properties d electrons in Fe and Ni.


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Collborators and References Reviews of DMFT: A. Georges G.

Kotliar W krauth and M . Rozenberg Rev Mod Pnys 68, 13 (1996). DMFT and LDA R. Chitra and G. Kotliar Phys. Rev. B 62,, 12715 (2000).

S. Savrasov and G. Kotliar cond-mat cond-mat 0106308.

DMFT study of Plutonium. S. Savrasov, G. Kotiar and E. Abrahams, Nature 410, 793 (2001). S. Savrasov and G. Kotliar

DMFT study of Iron and Nickel. S. Lichtenstein M Katsenelson and G. Kotliar Phys. Rev. Lett 87, (2001).


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Outline

Motivation: plutonium puzzles.

Review of Dynamical Mean Field Theory an Extension to realistic systems . DMFT and DFT.




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Case study in f electrons, Mott transition in the actinide series. B. Johanssen 1974 Smith and Kmetko Phase Diagram 1984.


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Pu: Complex Phase Diagram (J. Smith LANL)


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Small amounts of Ga stabilize the phase


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Problems with LDA

o DFT in the LDA or GGA is a well established tool for the calculation of ground state properties.

o Many studies (Freeman, Koelling 1972)APW methods

o ASA and FP-LMTO Soderlind et. Al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999) give

o an equilibrium volume of the an equilibrium volume of the phasephaseIs 35% lower than Is 35% lower than experimentexperiment

o This is the largest discrepancy ever known in DFT based calculations.


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Problems with LDA LSDA predicts magnetic long range

order which is not observed experimentally (Solovyev et.al.)

If one treats the f electrons as part of the core LDA overestimates the volume by 30%

LDA predicts correctly the volume of the phase of Pu, using full potential LMTO (Soderlind and Wills). This is usually taken as an indication that Pu is a weakly correlated system.


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Other Methods

LDA+ U (Savrasov and Kotliar Phys. Rev. Lett. 84, 3670, 2000, Bouchet et. al 2000) predicts correct volume of Pu with the constrained LDA estimate of U=4 ev. However, it predicts spurious magnetic long range order and a spectra which is very different from experiments.

Requires U=0 to treat the alpha phase, which has many physical properties in common with the delta phase.

Similar problems with the constrained (4 of the 5f electrons are treated as core ) LDA approach of Erikson and Wills.


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Conventional viewpoint

Alpha Pu is a simple metal, it can be described with LDA + correction. In contrast delta Pu is strongly correlated.

Constrained LDA approach (Erickson, Wills, Balatzki, Becker). In Alpha Pu, all the 5f electrons are treated as band like, while in Delta Pu, 4 5f electrons are band-like while one 5f electron is deloclized.

Same situation in LDA + U (Savrasov andGK Bouchet et. al. ) .Delta Pu has U=4,Alpha Pu has U =0.


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Problems with the conventional viewpoint of Pu

The specific heat of delta Pu, is only twice as big as that of alpha Pu.

The susceptibility of alpha Pu is in fact larger than that of delta Pu.

The resistivity of alpha Pu is comparable to that of delta Pu.

Only the structural and elastic properties are completely different.


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MAGNETIC SUSCEPTIBILITY


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Pu Specific Heat


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Anomalous ResistivityJ. Smith LANL


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Outline

Motivation: plutonium puzzles.

Review of Dynamical Mean Field Theory an Extension to realistic systems . DMFT and DFT.




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1

10

1( ) ( )

( )n nn k nk

G i ii t i

w ww m w

-

-é ùê ú= +Sê ú- + - Sê úë ûå

DMFT Impurity cavity construction: A. Georges, G. Kotliar, PRB, (1992)]

†

0 0 0

[ ] ( )[ ( , ')] ( ')o o o oS Go c Go c U n nb b b

s st t t t ¯= +òò ò

† †

, ,

( )( )ij ij i j j i i ii j i

t c c c c U n n

0

†( )( ) ( ) ( )L n o n o n S GG i c i c iw w w=- á ñ

10 ( ) ( )n n nG i i iw w m w- = + - D

0

1 † 10 0 ( )( )[ ] ( ) [ ( ) ( ) ]n n n n S Gi G G i c i c ia bw w w w- -S = + á ñ

Weiss field


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Single site DMFT, functional formulation

Express in terms of Weiss field (semicircularDOS)

The Mott transition as bifurcation point in functionals oG or F[], (G. Kotliar EPJB 99)

[ , ] log[ ]

( ) ( ) [ ]

ijn

n n

G Tr i t

Tr i G i G

w

w w

-GS =- - S -

S +F

[ ]DMFT atom ii

i

GF = FåLocal self energy (Muller

Hartman 89)

2

2

( )[ ] [ ]imp

iF T F

t

† †,

[ , ] ( ) ( ) ( )†[ ]locL f f f i i f i

impF Log df dfe


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Solving the DMFT equations

G0 G

I m p u r i t yS o l v e r

S .C .C .

•Wide variety of computational tools (QMC, NRG,ED….)

•Analytical Methods

G0 G

Im puritySo lver

S .C .C .


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DMFT

Construction is easily extended to states with broken translational spin and orbital order.

Large number of techniques for solving DMFT equations for a review see

A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]


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Schematic DMFT phase diagram one band Hubbard (half filling, semicircular DOS, role of partial frustration) Rozenberg et.al PRL (1995)


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Evolution of the Spectral Function with Temperature

Anomalous transfer of spectral weight connected to the proximity to an Ising Mott endpoint (Kotliar et.al.PRL 84, 5180 (2000))


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Localization Delocalization The Mott transition/crossover is driven by transfer of spectral weight from low to high energy as we approach the localized phaseControl parameters: doping, temperature,pressure… Intermediate U region is NOT perturbatively accessible. DMFT a new starting point to access this regime.


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Qualitative phase diagram in the U, T , plane (two band Kotliar and Rozenberg (2001))

Coexistence regions between localized and delocalized spectral functions.


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QMC calculationof n vs (Murthy Rozenberg and Kotliar 2001, 2 band, U=3.0)

diverges at generic Mott endpoints


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Combining LDA and DMFT

The light, SP (or SPD) electrons are extended, well described by LDA

The heavy, D (or F) electrons are localized,treat by DMFT.

LDA already contains an average interaction of the heavy electrons, substract this out by shifting the heavy level (double counting term)

The U matrix can be estimated from first principles or viewed as parameters


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Spectral Density Functional : effective action construction (Fukuda, Valiev and Fernando , Chitra and GK).

DFT, consider the exact free energy as a functional of an external potential. Express the free energy as a functional of the density by Legendre transformation. DFT(r)]

Introduce local orbitals, R(r-R)orbitals, and local GF

G(R,R)(i ) =

The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing sources for (r) and G and performing a Legendre transformation, (r),G(R,R)(i)]

' ( )* ( , ')( ) ( ')R Rdr dr r G r r i r


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Spectral Density Functional

The exact functional can be built in perturbation theory in the interaction (well defined diagrammatic rules )The functional can also be constructed expanding around the the atomic limit. No explicit expression exists.

DFT is useful because good approximations to the exact density functional DFT(r)] exist, e.g. LDA, GGA

A useful approximation to the exact functional can be constructed, the DMFT +LDA functional.


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LDA+DMFT functnl

2 *log[ / 2 ( ) ( )]

( ) ( ) ( ) ( )

1 ( ) ( ')( ) ( ) ' [ ]

2 | ' |

[ ]

R R

n

n KS

KS n n

i

LDAext xc

DC

R

Tr i V r r

V r r dr Tr i G i

r rV r r dr drdr E

r r

G

a b ba

w

w c c

r w w

r rr r

- +Ñ - - S -

- S +

+ + +-

F - F

åò

ò òå

Sum of local 2PI graphs with local U matrix and local

G1

[ ] ( 1)2DC G Un nF = - ( )0( ) i

ab

abi

n T G i ew

w+

= å

KS ab [ ( ) G V ( ) ]LDA DMFT a br r


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LDA+DMFT Self-Consistency loop

G0 G

Im puritySo lver

S .C .C .

0( ) ( , , ) i

i

r T G r r i e w

w

r w+

= å

2| ( ) | ( )k xc k LMTOV H ka ac r c- Ñ + =

DMFT

U

E

0( , , )HHi

HH

i

n T G r r i e w

w

w+

= å


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Comments on LDA+DMFT

• Static limit of the LDA+DMFT functional , with = HF reduces to LDA+U

• Removes inconsistencies and shortcomings of this approach. DMFT retain correlations effects in the absence of orbital ordering.

• Only in the orbitally ordered Hartree Fock limit, the Greens function of the heavy electrons is fully coherent

• Gives the local spectra and the total energy simultaneously, treating QP and H bands on the same footing.


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Outline






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Pu: DMFT total energy vs Volume Savrasov Kotliar Abrahams to appear in Nature


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Lda vs Exp Spectra

DO

S, s

t./[e

V*c

ell]


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Pu Spectra DMFT(Savrasov) EXP (Arko et.al)


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PU: ALPHA AND DELTA


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Dynamical Mean Field View of Pu(Savrasov Kotliar and Abrahams, Nature 2001)

Delta and Alpha Pu are both strongly correlated, the DMFT mean field free energy has a double well structure, for the same value of U. One where the f electron is a bit more localized (delta) than in the other (alpha).

Is the natural consequence of the model Hamiltonian phase diagram once the structure is about to vary.

This result resolves one of the basic paradoxes in the physics of Pu.


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Minimum of the melting point

Divergence of the compressibility at the Mott transition endpoint.

Rapid variation of the density of the solid as a function of pressure, in the localization delocalization crossover region.

Slow variation of the volume as a function of pressure in the liquid phase


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Minimum in melting curve and divergence of the compressibility at the Mott endpoint

( )dT V

dp S

Vsol

Vliq


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Cerium: melting T vs p


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Pu: Anomalous thermal expansion (J. Smith LANL)


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Double well structure and Pu Qualitative

explanation of negative thermal

expansion

Sensitivity to impurities which easily raise the energy of the -

like minimum.


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Double well structure and Pu

negative thermal expansion

Sensitivity to impurities which easily raise the energy of the -like

minimum.


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Future directions

Including short range correlations. Less local physics, C-DMFT.

Life without U, including the effects of long range Coulomb interactions, E-DMFT and GW.

Applications are just beginning, many surprises ahead……


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Outline




A case study with d electrons in Fe and Ni.


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Case study Fe and Ni

Archetypical itinerant ferromagnets

LSDA predicts correct low T moment

Band picture holds at low T Main puzzle: at high

temperatures has a Curie Weiss law with a moment much larger than the ordered moment.

Magnetic anisotropy


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Iron and Nickel: crossover to a real space picture at high T (Lichtenstein, Katsnelson and GK)


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Photoemission Spectra and Spin Autocorrelation: Fe (U=2, J=.9ev,T/Tc=.8) (Lichtenstein, Katsenelson,GK prl 2001)


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Photoemission and T/Tc=.8 Spin Autocorrelation: Ni (U=3, J=.9 ev)


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Iron and Nickel:magnetic properties (Lichtenstein, Katsenelson,GK PRL 01)

0 3( )q

Meff

T Tc

0 3( )q

Meff

T Tc


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Ni and Fe: theory vs exp

( T=.9 Tc)/ ordered moment

Fe 1.5 ( theory) 1.55 (expt) Ni .3 (theory) .35 (expt)

eff high T moment

Fe 3.1 (theory) 3.12 (expt)

Ni 1.5 (theory) 1.62 (expt)

Curie Temperature Tc

Fe 1900 ( theory) 1043(expt)

Ni 700 (theory) 631 (expt)


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Fe and Ni Satellite in minority band at 6 ev, 30

% reduction of bandwidth, exchange splitting reduction .3 ev

Spin wave stiffness controls the effects of spatial flucuations, it is about twice as large in Ni and in Fe

Mean field calculations using measured exchange constants(Kudrnovski Drachl PRB 2001) right Tc for Ni but overestimates Fe , RPA corrections reduce Tc of Ni by 10% and Tc of Fe by 50%.


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However not everything in low T phase is OK as far as LDA goes.. Magnetic anisotropy puzzle.

LDA predicts the incorrect easy axis(100) for Nickel .(instead of the correct one (111)

LDA Fermi surface has features which are not seen in DeHaas Van Alphen ( Lonzarich)

Use LDA+ U to tackle these refined issues, ( compare parameters with DMFT results )

Documents

Gabriel Kotliar Physics Department and Center for Materials Theory Rutgers University